On the Characterization and Selection of Diverse Conformational Ensembles with Applications to Flexible Docking

Abstract

To address challenging flexible docking problems, a number of docking algorithms pregenerate large collections of candidate conformers. To remove the redundancy from such ensembles, a central problem in this context is to report a selection of conformers maximizing some geometric diversity criterion. We make three contributions to this problem. First, we resort to geometric optimization so as to report selections maximizing the molecular volume or molecular surface area (MSA) of the selection. Greedy strategies are developed, together with approximation bounds. Second, to assess the efficacy of our algorithms, we investigate two conformer ensembles corresponding to a flexible loop of four protein complexes. By focusing on the MSA of the selection, we show that our strategy matches the MSA of standard selection methods, but resorting to a number of conformers between one and two orders of magnitude smaller. This observation is qualitatively explained using the Betti numbers of the union of balls of the selection. Finally, we replace the conformer selection problem in the context of multiple-copy flexible docking. On the aforementioned systems, we show that using the loops selected by our strategy can improve the result of the docking process.

Assessing the reconstruction of macromolecular assemblies with toleranced models

“…But the design of such models is a research topic in itself, as it is reminiscent from geometric covering theory: given a budget of (toleranced) balls, how can one best approximate a region of space representing a protein complex? Such questions are typically intractable (NP‐complete problems), although effective solutions have been found in related situations for parsimonious representations 10…”

“…First, a level set surface in the map may not represent the instances of that type accurately, in particular if the resolution of the map is not consistent across the model. Second, the task of segmenting a map into a prescribed number of regions (one per protein instance) is an ill‐posed problem if the number of local maxima of the map does not match the stoichiometry of that type and is another computationally hard problem related to the so‐called geometric max‐k‐cover problem otherwise 10…”

Computing the arrangement of circles on a sphere, with applications in structural biology

“…An obvious such strategy is the greedy one, which consists of incrementally selecting the s conformers: The conformer selected at the i-th stage is that yielding the best increment. These strategies are examined in [22], where the following theorems are proved:…”

“…Alternatively, one may skip the calculation of the covering lists, in which case the greedy selection can be carried out by recomputing for each candidate conformer C k the surface area of S j−1 ∪ C k , with S j−1 the selection obtained so far. In that case, one has [22]:…”

Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse‐Graining

“…Last but not least, our incentive to address approximation problems for balls comes from computational structural biology, whose ultimate goal is to unravel the relationship between the structure and the function of macro‐molecules. Originating with the work of Richards [LR71], molecular models represented as collections of van der Waals (vdW) balls and associated affine Voronoi diagrams have been instrumental to describe atomic packing properties [MJLC87, MLJ*87], to compute and decorate molecular surfaces [Con83, AE96], to exhibit correlations between structural and biological—bio‐physical properties of protein interfaces [BCRJ03, MDBC12], to select diverse conformational ensembles for mean field theory based docking algorithms [LSB*11], or to find entrance/exit passages to active sites [YFW*08].…”